A Linear-Correction Algorithm for Quasi-Synchronous DFT

Hindawi Mathematical Problems in Engineering Volume 2018, Article ID 1268905, 9 pages https://doi.org/10.1155/2018/1268905

Research Article A Linear-Correction Algorithm for Quasi-Synchronous DFT Zhongjun Fu

,1,2 Jianyu Wang

,2 Yun Ou,3 Genyuan Zhou,1 and Xiaorong Zhao1

1

School of Computer Engineering, Jiangsu University of Technology, 1801 ZHONGWU RD, Changzhou, China School of Automation, Nanjing University of Science and Technology, 200 XIAOLINGWEI, Nanjing, China 3 Changzhou Foreign School, 55 LIAOHE RD, Changzhou, China 2

Correspondence should be addressed to Jianyu Wang; jianyu [email protected] Received 15 August 2018; Revised 26 October 2018; Accepted 6 November 2018; Published 27 December 2018 Academic Editor: Ram´on I. Diego Copyright © 2018 Zhongjun Fu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Spectral leakage in the harmonic measured by quasi-synchronous DFT (QSDFT) is mainly due to short-range leakage caused by deviation in the signal frequency. By analysing the short-range-leakage characteristic of QSDFT, a linear-correction algorithm (LCQS) for QSDFT’s harmonic-analysis results is proposed. LCQS contains two linear-correction equations: an amplitudecorrection equation and an initial-phase-angle-correction equation. The former is constructed by the least-squares method, whereas the latter is generated based on the linear error characteristic of the QSDFT harmonic phase. Simulation and experimental results indicate that this proposed algorithm can efficiently increase the accuracy of the harmonic parameters over a wide frequency range by minimizing the short-range spectral leakage.

1. Introduction Harmonic analysis technology can break down complex signals into simple periodic ones to facilitate their understanding. This technique is widely used in many fields such as power-quality monitoring, electronic product inspection, and electrical equipment monitoring. The Fourier transform method, including discrete Fourier transforms (DFTs) and fast Fourier transforms (FFTs), is the most widely used method for harmonic analysis. However, two types of spectrum leakage—i.e., long-range leakage caused by the truncation effect (finite points sampling) and short-range leakage caused by the fence effect (asynchronous sampling)—appear when DFT and FFT are applied, resulting in inaccurate and untrustworthy analytical results [1–3]. Thus, many solutions have been presented, including the windowed-interpolation algorithm [4–7] and quasi-synchronous DFT (QSDFT) [8– 15]. The QSDFT algorithm is simple and easy to implement, and the measurement accuracy of the signal frequency is high when the frequency is deviated [3]. However, suppression of the short-range spectral leakage caused by signal-frequency deviation is insufficient, limiting analytical accuracy. Many improved methods to address this problem

have been presented in the literature, such as parameteradapted QSDFT [16], the periodic-sampling-points correction method [17], and the least-squares method [18]; however, none can completely suppress spectral leakage. [19] proposed the so-called variable picket fence (VPF) harmonic-analysis method, which effectively suppresses short-range spectrum leakage but requires two analytical operations to obtain results, making it unsuitable for real-time operation. QSDFT uses the multiple iterations integral method to suppress spectral leakage; however, it causes the spectral equation to become extremely complex, preventing a simple interpolation-correction function for the frequency deviation from being obtained. In this paper, a linear-correction algorithm (LCQS) for QSDFT is proposed, containing two equations: the amplitude-correction equation and the initialphase-angle-correction equation. It can easily correct the errors caused by frequency deviation. Simulation and application results show the validity of this algorithm.

2. QSDFT and VFP 2.1. QSDFT Algorithm. Consider a periodic signal, 𝑓(𝑡): ∞

𝑓 (𝑡) = 𝐴 0 + ∑ 𝐴 𝑘 sin (2𝜋𝑘𝑓1 𝑡 + 𝜑𝑘 ) 𝑘=1

(1)

2

Mathematical Problems in Engineering

Here, 𝑘 is the harmonic order, 𝐴 𝑘 and 𝜑𝑘 are the amplitude and initial phase of the kth harmonic, respectively, and 𝑓1 is the frequency of 𝑓(𝑡). According to the sampling frequency 𝑓𝑆 and the number of sampling points within a cycle 𝑁, uniformly spaced sampling is performed W+1 times in the range [𝑡0 , 𝑡0 +𝑊×𝑇𝑆 ] to obtain the sampled sequence 𝑓(𝑖), 𝑖 = 0 𝑡𝑜 𝑊. Here, 𝑊 is determined by the integral method, 𝑊 = 𝑛𝑁, when using trapezoidal integration methods; 𝑛 is the number of iterations. Appling Eqs. (2) and (3) to calculate the real part, 𝑎𝑘 and the imaginary part, 𝑏𝑘 , and thus the 𝐴 𝑘 and 𝜑𝑘 values of the kth harmonic, QSDFT is given by [8] 𝑎𝑘 =

𝑛 2𝐹𝑎𝑘

If the signal frequency does not deviate, 𝜇 = 1; if it does, 𝜇 ≠ 1 and Δ𝜇 = 𝜇 − 1

Suppose 𝑁 > (2 + 𝑁𝑓1 /𝑓S ), 𝑓S /𝑁 > 2𝑀|𝑓1 |, and 𝑛 is a large number; the following equations can be deduced from [20]: 𝑛

𝑛 𝐹𝑎1 = (𝜌11 ) 𝐴 1 cos (2𝜋𝑓1 𝑁𝑇𝑆 𝐶𝑛 + 𝜑1∗ ) 𝑛

𝑛 𝐹𝑏1 = (𝜌11 ) 𝐴 1 sin (2𝜋𝑓1 𝑁𝑇𝑆 𝐶𝑛 + 𝜑1∗ )

2 2𝜋 (𝑖) = ∑𝛾𝑗 𝑓 (𝑖 + 𝑗) cos (𝑘 𝑗) 𝑄 𝑗=0 𝑁

𝑛 𝑏𝑘 = 2𝐹𝑏𝑘 (𝑖) =

(2)

2 2𝜋 ∑𝛾 𝑓 (𝑖 + 𝑗) sin (𝑘 𝑗) 𝑄 𝑗=0 𝑗 𝑁

(3)

𝑎𝑘 ] 𝑏𝑘

𝐶𝑛 =

𝑖0 1 1 + 𝑛( − ) 𝑁 2 2𝑁

(7)

𝜌11 =

1 sin (𝜋𝑓Δ 𝑁𝑇𝑆 ) 𝑁 sin (𝜋𝑓Δ 𝑇𝑆 )

(8)

The fundamental initial phase 𝜑1 (𝑖) analyzed by QSDFT from the ith sample point is

𝐴 𝑘 = √𝑎𝑘2 + 𝑏𝑘2 𝜑𝑘 = 𝑡𝑔−1 [

𝜑1 (𝑖) = 𝑡𝑔−1 [

𝑛 𝐹𝑎1 (𝑖) ] 𝑛 𝐹𝑏1 (𝑖)

𝑖 1 1 = 2𝜋𝑓1 𝑁𝑇𝑆 [ 0 + 𝑛 ( − )] + 𝜑1∗ 𝑁 2 2𝑁

where 𝑖 is the initial sampling point, normally 𝑖 = 0; 𝛾𝑗 is the weighting coefficient, determined by the integral method, 𝑛 and 𝑁; and 𝑄 = ∑𝑊 𝑗=0 𝛾𝑗 is the sum of all weighting coefficients, with 𝑄 = (𝑁 + 1)𝑛 when 𝑊 = 𝑛𝑁. Strictly whole-cycle synchronous sampling is difficult to implement in the actual measurement process. Multiple factors, including the precision and integer-multiple error of the sampling clock and the signal-frequency deviation, will affect the sampling process, leading to short-range leakage of the discrete spectrum, as well as the most important source of leakage, namely, signal-frequency deviation. In theory, based on the convergence characteristics of multiple iterations, QSDFT can suppress short-range leakage caused by frequency deviation. However, the inhibitory effect upon the short-range leakage is not significant during practical applications. [19] pointed out that high analytical precision for the amplitude of QSDFT occurs only in the range of 50 ± 0.1Hz and that the initial-phase from QSDFT is untrustworthy except when it is close to the original frequency. In other words, QSDFT is almost completely incapable of suppressing short-range spectrum leakage.

Definition 1. The rate of signal-frequency deviation, 𝜇, is the degree of signal-frequency deviation, which is defined as follows: 𝑁𝑓1 𝑓𝑆

(4)

(9)

For two adjacent sampling points, the fundamental initial-phase difference is 𝜑1 (𝑖 + 1) − 𝜑1 (𝑖) =

2𝜋𝑓1 𝑓S

(10)

For engineering applications, the fundamental initial phases, 𝜑1(0) and 𝜑1(1), of the two adjacent sampling points can be calculated from two starting points 𝑖 = 0 and 𝑖 = 1 with Eq. (2), so 𝜇 is 𝜇=𝑁

𝑛 𝑛 𝑛 𝑛 𝑡𝑔−1 [𝐹𝑎1 (1) /𝐹𝑏1 (1)] − 𝑡𝑔−1 [𝐹𝑎1 (0) /𝐹𝑏1 (0)] 2𝜋

(11)

2.3. VFP Algorithm. [19] showed that when signal-frequency deviation occurs, the sampling position in the frequency domain is still 2𝜋/𝑁 (the corresponding analog frequency 𝑓𝑆 /𝑁), leading to great spectral leakage. Adjusting the frequency-domain sampling position to 𝜇2𝜋/𝑁 (the corresponding analog frequency 𝜇𝑓𝑆 /𝑁) will suppress short-range spectrum leakage. Thus, the improved VFP equation is 𝑛 𝑎𝑘 = 2𝐹𝑎𝑘 (𝑖) =

2.2. Rate of Signal-Frequency Deviation

𝜇=

(6)

Here, 𝑀 is the maximum harmonic order,

𝑊

𝑊

(5)

𝑏𝑘 =

𝑛 2𝐹𝑏𝑘

𝜇2𝜋 2 𝑊 𝑗) ∑𝛾𝑗 𝑓 (𝑖 + 𝑗) cos (𝑘 𝑄 𝑗=0 𝑁

𝜇2𝜋 2 𝑊 𝑗) (𝑖) = ∑𝛾𝑗 𝑓 (𝑖 + 𝑗) sin (𝑘 𝑄 𝑗=0 𝑁

(12)

However, the introduction of the frequency-deviation rate 𝜇 leads to two problems for the implementation of the VFP harmonic analysis algorithm:

Mathematical Problems in Engineering 1.00

3 A

A

0.90

B

B

0.80

Amplitude

0.70 0.60

C

C

0.50 0.40 0.30 0.20

95

100

90

85

80

75

70

65

60

55

50

45

40

35

30

25

20

15

5

10

0.00

0

0.10

−30 −25 −20 −15 −10 −5

0

5

10 15 20 25 30

Deviation (Hz)

Frequency (Hz) 40Hz 50Hz 55Hz

Figure 1: The discrete amplitude spectrum of QSDFT with signal frequency.

(1) the sine/cosine calculation of the Fourier transform factor cannot be achieved using the look-up-table method, and the computational cost increases significantly; (2) the VFP harmonic calculation must be completed after the sample is completed and cannot be calculated step by step.

3. Linear-Correction Algorithm 3.1. Amplitude Correction. Because the spectral function of QSDFT is too complex, it is almost impossible to derive a simple interpolation iterative formula to correct the analytical error caused by frequency deviation. This paper gives a linear amplitude-correction method based on the principle of least squares, the specific idea of which is (1) to construct the error curve between the fundamental amplitude error and the rate of signal-frequency deviation 𝜇 according to the law relating these quantities; (2) to obtain an equation for the error curve between the fundamental amplitude and 𝜇 using the least-squares method, as well as the linear-correction equation between the fundamental amplitude and 𝜇; (3) to extend the linear-correction equation of fundamental amplitude to each harmonic and obtain such an equation between each harmonic amplitude and 𝜇. We know that if the signal frequency 𝑓1 deviates 𝜇 times to 𝜇𝑓1 , the spectrum peak of the QSDFT amplitude will stretch 𝜇 times along with the frequency axis to 𝜇𝑓𝑠 /𝑁, and amplitudeanalysis error will result from it if the peak is still found at 𝑓𝑠 /𝑁. For example, there are three fundamental signals of different frequencies with the same amplitude of 1: the ideal signal frequency of 50 Hz, and the deviated frequencies at 40 Hz and 55 Hz; using QSDFT to analyze the harmonics of

these three signals, we find that they have peaks of amplitudes A, B, and C, respectively (Figure 1, left). Obviously, only the fundamental amplitude of 50 Hz obtained by QSDFT is correct, and the obtained fundamental amplitudes of the other two signals have larger errors. Figure 1 suggests a means of obtaining the fundamental amplitude-error curve of QSDFT. This involves recording the fundamental amplitude-analysis results of these three signals, together with their frequency deviations (0Hz, +5Hz, and 10 Hz, resp.). Then, these deviations can be used as abscissas (in units of hertz) and the fundamental amplitude can be used as ordinates; points A, B, and C can then be plotted as Figure 1, right. Then, the fundamental signal from 20 Hz to 80 Hz will be continuously generated and QSDFT harmonic analysis will be carried out. The fundamental amplitude is recorded and plotted, and its error curve can be obtained (Figure 1, right). The equation of the QSDFT fundamental amplitude-error curve is 𝑎1 (Δ𝜇) =

2𝜋 (Δ𝜇 + 1) 2 𝑊 2𝜋 𝑗) cos ( 𝑗) ∑𝛾 sin ( 𝑄 𝑗=0 𝑗 𝑁 𝑁

2𝜋 (Δ𝜇 + 1) 2 𝑊 2𝜋 𝑏1 (Δ𝜇) = ∑𝛾𝑗 sin ( 𝑗) sin ( 𝑗) 𝑄 𝑗=0 𝑁 𝑁 𝐴 1 (Δ𝜇) = √𝑎12 (Δ𝜇) + 𝑏12 (Δ𝜇)

(13)

(14)

The least-squares principle is employed to obtain the solution of Eq. (14); set 𝑀

𝑖

𝐴 1 (Δ𝜇) = ∑𝛽𝑖 (Δ𝜇) 𝑖=0

(15)

4

Mathematical Problems in Engineering

There are two unknown quantities, 𝑀 and 𝛽𝑖 , in Eq. (15) when the error curve of the fundamental amplitude is known. Assuming that 𝑀 is known, set 𝛽0 [ ] [ 𝛽1 ] [ ] [ . ] [ . ] [ . ] ] 𝐵=[ [𝛽 ] [ 𝑖] [ ] [ . ] [ .. ] [ ]

(16)

In a certain range searching 𝑀 to make 𝐺 minimum, we can obtain the value of 𝑀. The number of iterations, n, is different, and the fundamental amplitude-error curve is not consistent. Table 1 shows 𝛽𝑖 with different 𝑛. We also know that the spectral peak of the kth harmonic amplitude will deviate by 𝑘Δ𝜇 from the ideal position when the signal frequency deviates by Δ𝜇, so the amplitude linearcorrection function of QSDFT is 2

𝐴𝑘 =

[𝛽𝑀] Δ𝑘 = [1 Δ𝜇𝑘 ⋅ ⋅ ⋅ (Δ𝜇𝑘 )𝑖 ⋅ ⋅ ⋅ (Δ𝜇𝑘 )𝑀]

(17)

𝐴 1𝑘 = 𝐴 1 (Δ𝜇𝑘 ) = Δ𝑘 𝐵

(18)

Thus, We apply Eqs. (13) and (14) to produce 𝐿-group data 𝐴 1𝑘 and Δ 𝑘 , denoted as 𝐴 10

[ ] [𝐴 11 ] [ ] [ . ] [ . ] [ . ] ] 𝐴=[ [𝐴 ] [ 1𝑘 ] [ ] [ . ] [ .. ] [ ] [𝐴 1𝐿 ] Δ0

𝐴 1 (𝑘Δ𝜇)

𝑖

𝑀

1 Δ𝜇0 ⋅ ⋅ ⋅ (Δ𝜇0 ) ⋅ ⋅ ⋅ (Δ𝜇0 )

[ [ ] [ [Δ1 ] [1 Δ𝜇1 [ ] [ [ . ] [. . [ . ] [. . [ . ] [. . ] [ Δ=[ [Δ ] = [ [ 𝑘 ] [1 Δ𝜇𝑘 [ ] [ [ . ] [. . [ .. ] [ . . [ ] [. . [ [Δ𝐿 ] [1 Δ𝜇𝐿

] 𝑖 𝑀] ⋅ ⋅ ⋅ (Δ𝜇1 ) ⋅ ⋅ ⋅ (Δ𝜇1 ) ] ] .. .. ] ] d . d . ] ] 𝑖 𝑀] ⋅ ⋅ ⋅ (Δ𝜇𝑘 ) ⋅ ⋅ ⋅ (Δ𝜇𝑘 ) ] ] .. .. ] ] d . d . ] ] 𝑖 𝑀 ⋅ ⋅ ⋅ (Δ𝜇𝐿 ) ⋅ ⋅ ⋅ (Δ𝜇𝐿 ) ]

𝜑𝑘 = 𝑡𝑔−1 [

𝑛 𝐹𝑎𝑘 ] = 2𝜋𝑓1 𝑁𝑇𝑆 𝐶𝑛 + 𝜑𝑘∗ 𝑛 𝐹𝑏𝑘

(26)

𝑛 𝐹𝑎𝑘 𝑛 ) ≈ 𝜑𝑘 + 𝐶𝑘 𝐹𝑏𝑘

(27)

So,

The least-squares solution of 𝐵 is −1

(22)

The remaining issue is to determine the unknown quantity 𝑀. Suppose the error between the fitted value and the true value is 𝛿𝑘 = 𝐴 1𝑘 − Δ𝑘 𝐵

(23)

The mean square error of 𝛿𝑘 is 𝐿

2

𝐿

𝑘=0

𝑛 𝐹𝑘𝑎 − 𝐶𝑘 𝑛 ) 𝐹𝑘𝑏

(28)

3.3. Analytical Process of LCQS. The analytical process of LCQS consists of five steps: (1) sampling 𝑊+2 data. Adding one more sampling point is necessary for LCQS due to the need for two starting points, 𝑖 = 0 and 𝑖 = 1; 𝑛 𝑛 𝑛 𝑛 (0), 𝐹𝑏1 (0), 𝐹𝑎1 (1), and 𝐹𝑏1 (1) for 𝑖 = 0 (2) calculating 𝐹𝑎1 and 𝑖 = 1, respectively, using Eq. (2); (3) calculating 𝜇 and Δ𝜇 by Eqs. (11) and (5); (4) calculating the raw amplitude and the initial phase of each harmonic by Eqs. (2) and (3); (5) correcting the amplitude and the initial phase of each harmonic using Eqs. (25) and (28), respectively.

4. Simulation 2

𝐺 = √ ∑ 𝛿𝑘 = √ ∑ (𝐴 1𝑘 − Δ𝑘 𝐵) 𝑘=0

𝜑𝑘 = 𝑡𝑔−1 ( where 𝐶𝑘 = 𝑛(𝑘Δ𝜇)𝜋.

(21)

𝐵 = (Δ𝑇 Δ) Δ𝑇 𝐴

Here, 𝐶𝑘 is a constant. Eq. (27) suggests that there is a linear relationship between the analytical results and the real values of the harmonic initial-phase-angle of signal-frequency deviation. However, no method for determining 𝐶𝑘 is given in [20]. A large number of simulation experiments show that 𝐶𝑘 is associated with the rate of signal-frequency deviation 𝜇, the number of iterations 𝑛, and the harmonic order 𝑘 and is not affected by the number of sampling points 𝑁 and other parameters. Thus, the initial-phase-angle linear-correction function of QSDFT is

(20)

Then, 𝐴 = Δ𝐵

(25)

3.2. Initial-Phase Angle Correction. Suppose 𝑁 > (2 + 𝑁𝑓1 /𝑓S ), 𝑓S /𝑁 > 2𝑀|𝑓1 |, and 𝑛 is a large number; the following equation can be deduced from [20]:

𝜑𝑘𝑛 = 𝑡𝑔−1 ( (19)

2

𝑛 𝑛 ) + (𝐹𝑘𝑏 ) 2√(𝐹𝑘𝑎

(24)

We design two groups of simulation experiments to examine the effectiveness of LCQS: an analytical-accuracy experiment

𝛽0 𝛽1 𝛽2 𝛽3 𝛽4 𝛽5 𝛽6 𝛽7

𝑛=5 1 -4.72E-8 -8.35318 -0.00017 32.09637 -0.05003 -75.5957 0.17132

𝑛=6 1 -3.2E-10 -10.0238 -2.71E-6 46.88846 -0.00260 -136.391 0.11315

𝑛=7 1 1.36E-11 -11.6945 -4.54E-8 64.47217 -7.20E-5 -223.338 -0.00900

𝑛=8 1 1.83E-11 -13.3651 -8.80E-9 84.84648 -4.43E-7 -341.029 -0.00079 𝛽8 𝛽9 𝛽10 𝛽11 𝛽12 𝛽13 𝛽14 𝛽15

Table 1: Values of 𝛽𝑖 with different 𝑛. 𝑛=5 122.6894 0.11220 -144.047 -1.45938 108.3661 2.51764

𝑛=6 276.8987 -0.73669 -407.863 2.12560 373.5219 -2.75833

𝑛=7 546.7441 0.00335 -1005.52 0.36084 1400.512 -1.69420 -1226.43 2.65455

𝑛=8 975.7476 0.03638 -2107.86 -0.28833 3430.099 1.07551 -3409.06 -1.90542

Mathematical Problems in Engineering 5

6

Mathematical Problems in Engineering Error (%)

Error (%) 3.0E+01

1.0E-03 8.0E-04 6.0E-04 4.0E-04 2.0E-04 0.0E+00 -2.0E-04 -4.0E-04 -6.0E-04 -8.0E-04 -1.0E-03

2.5E+01 2.0E+01 1.5E+01

5

7

9

Harmonic Order

11

13

15

Frequency (Hz)

0.0E+00 1

3

5

7

9

Harmonic Order

11

(a)

15

Frequency (Hz)

(b)

Error (%) 0.0E+00 -5.0E-08 -1.0E-07 -1.5E-07 -2.0E-07

3

5

7

Harmonic Order

9

11

13

49.5 49.6 49.7 49.8 49.9 50 50.1 50.2 50.3 50.4 50.5

Error (%) 1.0E-12 5.0E-13 0.0E+00 -5.0E-13 -1.0E-12 -1.5E-12 -2.0E-12 -2.5E-12 -3.0E-12 -3.5E-12 -4.0E-12 1

13

49.5 49.6 49.7 49.8 49.9 50 50.1 50.2 50.3 50.4 50.5

3

5.0E+00

15

Frequency (Hz)

(c)

-2.5E-07 -3.0E-07 1

3

5

7

Harmonic Order

9

11

13

49.5 49.6 49.7 49.8 49.9 50 50.1 50.2 50.3 50.4 50.5

1

49.5 49.6 49.7 49.8 49.9 50 50.1 50.2 50.3 50.4 50.5

1.0E+01

15

Frequency (Hz)

(d)

Figure 2: The amplitude errors of these four methods. (a) Hanning, (b) QSDFT, (c) VFP, and (d) LCQS.

and a time complexity experiment. For comparison, we also perform the same experiments with the Hanning interpolation [21], QSDFT, and VFP methods. 4.1. Analytical-Accuracy Experiment. Some waveform signals are generated from Eq. (29) to verify the analytical accuracy of these four methods. The signals are measured with 𝑓S = 6,400 𝐻𝑧, 𝑁 = 128, and the number of iterations 𝑛 = 8: 8

𝑓 (𝑡) = ∑ 𝐴 2𝑘−1 sin [(2𝑘 − 1) 2𝜋𝑓1 𝑡 + 𝜑2𝑘−1 ]

(29)

𝑘=1

Here, 𝐴 2𝑘−1 and 𝜑2𝑘−1 are arbitrary; 𝑓1 = 49.5 to 50.5Hz; Amplitude Relative Error = (Analyzed-Theoretical)/Theoretical ∗ 100%; Initial-phase Absolute Error = (AnalyzedTheoretical)∘ . Figure 2 shows the relative errors in the amplitude obtained by each of the four methods. In Figure 2, the x-axis is the signal frequency in Hz; the Y-axis is the harmonic order from 1 to 15; and the z-axis is the relative percent error in amplitude. As can be seen from Figure 2, the error of QSDFT

is the largest and reaches 10% level; the error of Hanning interpolation is second and is to 10−4 % level; the error of LCQS is to 10−7 % level; and the error of VFP is minimal at 10−12 % level. Figure 3 illustrates the absolute error of the initial phase obtained by the four above-mentioned methods. In Figure 3, the x-axis is the signal frequency in hertz, the y-axis is the harmonic order from 1 to 15, and the z-axis is the absolute error of the initial-phase angle in degrees. As can be seen from Figure 3, the error in QSDFT is largest, reaching the 102 degree level; however, if 360-degree periodicity is considered, the error surface falls in a plane, which is also the theoretical basis for LCQS; the error of Hanning interpolation is second largest, reaching the 10−2 -degree level; the error of LCQS reaches the 10−8 -degree level, and that of VFP reaches the 10−9 -degree level. However, the error of LCQS only slightly protrudes at the frequency and harmonic edges, and the overall error falls almost in a plane. The analytical-accuracy experiments show that (1) LCQS makes up for the short-range leakage of the QSDFT algorithm, relaxes the requirement for synchronous

Mathematical Problems in Engineering

7

Error (∘ )

Error (∘ )

9.0E-02 8.0E-02 7.0E-02 6.0E-02 5.0E-02 4.0E-02 3.0E-02 2.0E-02 1.0E-02 0.0E+00

2.0E+02 1.5E+02 1.0E+02 5.0E+01 0.0E+00 -5.0E+01

5

7

9

Harmonic Order

11

13

15

-2.0E+02 1

Frequency (Hz)

3

5

7

9

Harmonic Order

11

(a)

6.0E-09 5.0E-09 4.0E-09 3.0E-09 2.0E-09

1

3

5

7

Harmonic Order

9

11

13

49.5 49.6 49.7 49.8 49.9 50 50.1 50.2 50.3 50.4 50.5

1.0E-09 -1.0E-09

Frequency (Hz)

13

15

(b)

Error (∘ ) 7.0E-09

0.0E+00

49.5 49.6 49.7 49.8 49.9 50 50.1 50.2 50.3 50.4 50.5

3

-1.5E+02

Error (∘ ) 2.5E-08 2.0E-08 1.5E-08 1.0E-08 5.0E-09 0.0E+00 -5.0E-09 -1.0E-08 -1.5E-08 -2.0E-08 -2.5E-08 -3.0E-08 1

3

5

Frequency (Hz)

7

Harmonic Order

15

(c)

9

11

13

49.5 49.6 49.7 49.8 49.9 50 50.1 50.2 50.3 50.4 50.5

1

49.5 49.6 49.7 49.8 49.9 50 50.1 50.2 50.3 50.4 50.5

-1.0E+02

Frequency (Hz)

15

(d)

Figure 3: The initial-phase errors of these four methods. (a) Hanning, (b) QSDFT, (c) VFP, and (d) LCQS.

sampling, and significantly increases the analytical accuracy of each harmonic. (2) Compared with QSDFT, LCQS can obtain high analytical precision simply by adding one sample point and one fundamental initial-phase calculation. (3) Compared with the VFP, LCQS greatly reduces the complexity of the algorithm and can be calculated step by step. 4.2. Time Complexity Experiment. In order to qualitatively describe the time complexity of the four harmonic-analysis methods, we designed a simulation experiment. A waveform signal is generated from Eq. (29), where 𝐴 2𝑘−1 and 𝜑2𝑘−1 are arbitrary, 𝑓1 = 50Hz, 𝑓S = 6,400𝐻𝑧, 𝑁 = 128, and the number of iterations is 𝑛 = 8. In the MATLAB R2016b software environment, the analysis times of these four methods are recorded. To exclude other interference factors, the same 200 consecutive analytical experiments were performed, and the time average was taken. The experimental results are presented in Table 2. The simulation results show that the time complexity of LCQS is only slightly higher than that of the QSDFT at

Table 2: The required time (s) for the four harmonic-analysis algorithms for different numbers of iterations. 𝑛 4 5 6 7 8

Hanning 0.001826 0.002388 0.002684 0.003119 0.003519

QSDFT 0.001139 0.001440 0.001729 0.001979 0.002218

VFP 0.010694 0.013180 0.015721 0.017495 0.020051

LCQS 0.001844 0.002208 0.002438 0.002757 0.002998

different iterations, but about an order of magnitude lower than the VFP algorithm.

5. Experimental Results To evaluate the practical performance of LCQS, we embedded LCQS into a measuring device, JCQ-5 (Figure 4(a)), designed to monitor the capacitive equipment online. This equipment is important to the power grid and usually requires online

8

Mathematical Problems in Engineering Table 3: Experimental results with different methods. 𝛿/∘

𝐴 𝑖1 /%

𝑓

Hanning -0.5000 -0.2667 0.0667 -0.1000 0.4333

49.5 49.7 50.0 50.3 50.5

QSDFT -1.3333 -0.5000 0.3333 -0.6667 -1.4333

u i

C

iC

VPF -0.1667 -0.0333 0.0667 -0.0667 -0.1333

Hanning -0.079 -0.035 0.018 -0.028 0.083

QSDFT -0.020 -0.017 0.008 0.012 0.017

VPF 0.005 -0.002 0.002 0.003 -0.006

LCQS -0.006 -0.002 0.001 -0.003 0.006

i

iC

Capacitive Equipment

LCQS -0.2000 -0.0667 0.1000 -0.1000 -0.1667

R iR 



iR u

(a)

(b)

Figure 4: Harmonic analysis by JCQ-5. (a) Measuring model of the capacitive equipment, (b) JCQ-5 measuring the signal from JB706.

monitoring of its current, 𝑖, and dissipation angle, 𝛿, or dissipation factor, tan 𝛿, to evaluate its status (Figure 4(a)): 𝛿=

𝜋 − (𝜑𝑖1 − 𝜑𝑢1 ) 2

(30)

Here, 𝜑𝑖1 and 𝜑𝑢1 are the fundamental initial phases of 𝑖 and 𝑢, respectively. JCQ-5 consists of four parts: Voltage / Current Transformer, Signal Conditioning, Synchronous ADC AD7606, and a CPU STM32F103. JCQ-5 samples 𝑖 and 𝑢 synchronously; in this device, we set 𝑓S = 6,400𝐻𝑧, 𝑁 = 128, and 𝑛 = 8. It calculates 𝐴 𝑖1 (fundamental amplitude of 𝑖), 𝜑𝑖1 , 𝜑𝑢1 , 𝛿 and tan 𝛿. A highly precise and stable threephase harmonic source JB706 (Figure 4(b)) was used as the signal source. The output parameters of JB706 were as follows: 𝐴 𝑢1 = 81.6𝑉, 𝜑𝑢1 = 0∘ , 𝐴 𝑖1 = 0.03𝐴, 𝜑𝑖1 = 85∘ , 𝑓𝑢1 = 𝑓𝑖1 = 49.5 ∼ 50.5𝐻𝑧. The amplitude of the third and fifth harmonics was 5% and 1% of that of the fundamental, respectively. For comparison, we also embedded the Hanning interpolation, QSDFT, and VPF into JCQ-5; their performances are shown in Table 3. Among them, the error of 𝐴 𝑖1 is relative error, the unit is %; the error of 𝛿 is absolute error, and the unit is degree. For the measurement accuracy of 𝐴 𝑖1 , QSDFT is inferior to the other methods, while VPF and LCQS have the same level of high accuracy. For the measurement accuracy of 𝛿, Hanning has a relatively poor result because of its lower

initial-phase analysis accuracy; QSDFT can also obtain the correct result, but only because the errors of 𝜑𝑢1 and 𝜑𝑖1 cancel each other out and the measurement results of 𝜑𝑢1 and 𝜑𝑖1 are not correct; VPF and LCQS still have the same high level of accuracy.

6. Conclusions This paper proposed a linear-correction algorithm (LCQS) for QSDFT; it has the following features: (1) Compared to QSDFT, LCQS suppresses the shortrange spectral leakage efficiently; compared to VPF, LCQS reduces the computations and can be operated in real time. (2) LCQS loosens the requirements for synchronous sampling and increases the analytical accuracy of amplitude and initial-phase determination significantly. (3) LCQS truly implements the linear correction of QSDFT harmonic-analysis results when the signal frequency deviates. (4) LCQS can work in real time and can be easily transplanted into existing hardware systems, making it a very practical algorithm.

Data Availability The data used to support the findings of this study are included within the article.

Mathematical Problems in Engineering

9

Conflicts of Interest The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

[14]

[15]

This study has been supported by the National Natural Science Foundations of China under Grant no. 61472166.

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